Question

given the function, answer the following questions below.\\( f(x) = 31(x - 5)^2 - 124 \\)\
what are the solutions to \\( x \\)?\
\\( x = \\)\
what are the coordinate point(s) of the \\( x \\)-intercept(s)?\
if there are no \\( x \\)-intercepts, put dne in the answer box. do not use decimals.\
question help: \\( \boxed{\text{video}} \\)

Explanation:

Step1: Find x-intercepts (set f(x)=0)

To find the x - intercepts, we set \( f(x) = 0 \). So we have the equation:

$$ 31(x - 5)^2-124=0 $$

Step2: Isolate the squared term

First, add 124 to both sides of the equation:

$$ 31(x - 5)^2=124 $$

Then, divide both sides by 31:

$$ (x - 5)^2=\frac{124}{31} $$

Since \( \frac{124}{31} = 4 \), the equation becomes:

$$ (x - 5)^2 = 4 $$

Step3: Take square root of both sides

Take the square root of both sides. Remember that when we take the square root of a number, we get both a positive and a negative solution. So:

$$ x - 5=\pm\sqrt{4} $$

Since \( \sqrt{4}=2 \), we have:

$$ x - 5=\pm2 $$

Step4: Solve for x

Case 1: When \( x - 5 = 2 \), add 5 to both sides:

$$ x=2 + 5=7 $$

Case 2: When \( x - 5=- 2 \), add 5 to both sides:

$$ x=-2 + 5 = 3 $$

Step5: Find x - intercept coordinates

The x - intercepts occur where \( y = 0 \) (since \( f(x)=y \)). So the coordinate points of the x - intercepts are \( (3,0) \) and \( (7,0) \)

Answer:

The solutions for \( x \) are \( x = 3 \) and \( x = 7 \). The coordinate points of the x - intercepts are \( (3,0) \) and \( (7,0) \)